Dynamical Lehmer's conjecture

Let $f\in\mathbb{Z}[z]$ be a fixed monic polynomial of degree $d\geq 2$, and let $\mathrm{m}_f(P)$ denote the dynamical Mahler measure of a single-variable polynomial $P\in\mathbb{Z}[x]$. Dynamical Lehmer's conjecture. There is some $\delta=\delta_f>0$ such that any single-variable polynomial $P\in\mathbb{Z}[x]$ with $\mathrm{m}_f(P)>0$ satisfies

$$\mathrm{m}_f(P)>\delta.$$

This is the dynamical analogue of Lehmer's question about whether positive Mahler measures of integer polynomials are bounded away from zero. The source attributes this conjecture to Silverman; its resolution is not specified here.